You Can Approximate Pi by Dropping Needles on the Floor

Happy Pi Day! March 14 is the date that otherwise rational people celebrate this irrational number, because 3/14 contains the first three digits of pi. And hey, pi deserves a day. By definition, it’s the ratio of the circumference and diameter of a circle, but it shows up in all kinds of places that seem to have nothing to do with circles, from music to quantum mechanics.

Pi is an infinitely long decimal number that never repeats. How do we know? Well, humans have calculated it to 314 trillion decimal places and didn’t reach the end. At that point, I’m inclined to accept it. I mean, NASA uses only the first 15 decimal places for navigating spacecraft, and that’s more than enough for earthly applications.

The coolest thing, for me, is that there are many ways to approximate that value, which I’ve written about in the past. For instance, you can do it by oscillating a mass on a spring. But maybe the craziest method of all was proven in 1777 by George Louis Leclerc, Comte de Buffon.

Decades earlier, Buffon had posed this as a probability question in geometry: Imagine you have a floor with parallel lines separated by a distance d. Onto this floor, you drop a bunch of needles with length L. What is the probability that a needle will cross one of the parallel lines?

A picture will help you understand what’s happening. Let’s say I drop just two needles on the floor (feel free to replace the needles with something safer, like toothpicks). Also, just to make things easier later, we can say that the needle length and line spacing are equal (d = L).

You can see that one of the needles crosses a line and the other doesn’t. OK, but what are the chances? This is not the most trivial problem, but let’s think about just one dropped needle. We only care about two values—the distance (x) from the farther end of the needle to a line, and the angle of the needle (θ) with respect to a perpendicular (see the diagram below). If x is less than half the spacing between lines, we get a needle-crossing. As you can see, you’d get a higher probability with a smaller x or a smaller θ.